Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

摘要

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems (x) over dot (t) = A(0)x(t) + A(1)x(t - tau) + B(0)u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can , be expressed as a linear system of equations, whose unknown is the value U(tau /2) is an element of R-nxn, i.e., the delay Lyapunov matrix at time tau/2. This linear matrix equation with n(2) unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n(2) x n(2) matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation MX +/- X-T N=C, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n approximate to 1000, i.e., a linear system with n(2) approximate to 10(6) unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.